Tottenham, who sacked boss Andre Villas-Boas on Monday following their crushing 5-0 home defeat by Liverpool, looked on course for victory when forgotten man Emmanuel Adebayor volleyed them ahead.
But the north Londoners’ defensive frailties were exposed again as Matt Jarvis equalised with 10 minutes left and Modibo Maiga headed the winner shortly afterwards.
“We saw the shaken nerve ends of Tottenham when we scored. We then went for the jugular,” West Ham manager Sam Allardyce said on Sky Sports.
In the night’s other quarter-final, Ashley Young scored his first goal in 18 months as champions Manchester United reached the last four with a 2-0 win at Stoke City in a match held up for 10 minutes by a hail storm in the first half.
West Ham’s reward was a two-legged semi-final against Manchester City, who booked their place in the last four on Tuesday, and Manchester United will play Sunderland, who shocked Chelsea in their quarter-final.
Amid grumblings about Tottenham’s sterile attacking performances and stolid tempo, caretaker manager Tim Sherwood, touted by many as a potential long-term appointment, sought to make an immediate impression.
He cast aside Villas-Boas’s dogged commitment to playing with a solitary striker and brought Adebayor back into the fold for his first start of the season.
They began with a spring in their step and laid siege to the West Ham goal in the opening stages, hemming the visitors into their own half and creating a string of good chances.
The script seemed to have been written when Adebayor put the hosts ahead in emphatic fashion, arriving late on to Jermain Defoe’s cross to smash home a volley off the underside of the bar.
It proved a false dawn, however, as West Ham levelled when Jarvis slammed the ball high into the roof of the net after Matt Taylor had flicked on a long pass forward and Maiga thumped home a fantastic low header from Mohamed Diame’s cross.
“We started the game really well. We created chances in the first 20 minutes, but couldn’t make the breakthrough. Later we gave it away and encouraged West Ham – they got the breaks in the end,” said Tottenham’s caretaker boss.
“Give them credit – they bombarded us and stuck to their task.”
England international Young was handed a rare United start with question marks hanging over his future at Old Trafford, but he reminded fans of his talent with a pile-driver of a shot from 20 metres to put the visitors ahead in the second half.
Left back Patrice Evra curled home a cool finish, clipping the ball into the far corner, to wrap up victory for David Moyes’s team with 12 minutes remaining.
“I’m glad we’re through. It will be a tough semi-final against Sunderland. Sunderland played well to beat Chelsea so anything can happen but we are really pleased to be in the last four,” Moyes told Sky Sports.
“The conditions were a leveller and it was really difficult to play in tonight. It’s some of the worst conditions I’ve played or managed in. But once we got the first goal we felt confident.
“This club is used to winning trophies, it is business as usual.”
(Editing by Ed Osmond)